A random vibration is a motion which is non-deterministic and which cannot be accurately predicted in a deterministic manner. Instead, a random vibration can be characterised statistically (e.g. based on historic data) and statistical techniques can be used to predict the motion of the random vibration (e.g. the probability of a particular acceleration and displacement magnitude in a particular time-period). A random vibration in a structure is typically induced by a random load (e.g. an acoustic pressure load or a mass subjected to inertia effects under random oscillation). The response of a structure subject to a random load is generally characterised in terms of root-mean-square quantities (such as stress or displacement), which provide a mean value of the particular random quantity over time.
A random load typically comprises all frequencies within a given frequency range at all times. Accordingly, when a random load is applied to a structure, all resonance modes of the structure within the frequency range of the random load will be excited at the same time and random vibration will result. Consequently, a fatigue analysis of a structure under random vibration is different to a fatigue analysis of the structure under harmonic excitation.
The stress induced in a structure due to random vibration is a useful parameter for assessing the strength and fatigue endurance of the structure. Assessments of this kind are important in a number of industries, including the aviation and automotive industries. For example, fatigue in airframe structures due to acoustic loading is a particular safety concern in the aviation industry, and airworthiness authorities typically require that airframe structures meet prescribed minimum standards, as determined by computational analysis, experimental testing and aircraft service history.
In order to address the above requirements, there has been much effort to develop methods that can accurately predict the stresses in structures induced by random vibration. However, current methods remain either prohibitively expensive (in a computational sense) or incapable of providing the required accuracy. Thus, additional experimental evidence is normally required before the structure in question can be deployed in an operational environment.
For simple structures, a forced response analysis is a practical analysis tool for determining the stress in a structure due to random vibration. However, the direct solution (or closed-form solution) for a forced response analysis is confined to simple cases such as single-degree-of-freedom structures. In reality, most structures behave as multiple-degrees-of-freedom systems.
More recently, numerical methods such as the Finite Element Method (FEM) have been used to model the stresses in structures induced by random loading. FEM is a numerical technique for finding approximate solutions to partial differential equations (PDEs). The method involves discretisation of the structure (or “domain”) into a plurality of elements (the “mesh”) defined by points in space (“nodes”). For each element in the domain, the relevant PDEs to be solved can be approximated using linear functions. The result of the discretisation process for all elements in the domain is a large dimensional linear problem of finite dimension, the solution of which will approximately solve the original PDEs. A skilled person would be well versed in implementation of FEM and its use, and further details will not be discussed herein.
With the advent of increased computing speeds, software codes employing FEM, such as NASTRAN™ (as implemented in MSC NASTRAN™ by MSC Software Corporation, Santa Ana, Calif., USA) have been used to perform direct random response analyses of complex structures subjected to a random loading. However, such analyses involve direct solution of the equations of motion for the structure in question over the range of frequencies defining the random loading. Such analyses are complex and the required computational power is often prohibitive, thereby limiting their application to simple structures comprising relatively few degrees of freedom.
Blevins [1] demonstrated a method for calculating the response of a structure due to forced vibration, based on a normal modal solution obtained from finite element analysis. Blevins demonstrated that an N by N multiple-degrees-of-freedom system could be transformed to N single-degree-of-freedom systems; each single-degree-of-freedom system being solvable using a direct (or closed-form) solution.
Conventionally, the deflection of a single-degree-of-freedom system can be solved in exact form by applying a force (either harmonic or random oscillating) to a mass, spring and damping system. In such systems, the deflection is proportional to stress, which itself is dependent on the applied boundary conditions (e.g. clamped or supported edges). The maximum stress can be calculated by scaling the maximum deflection obtained from the modal solution for the structure. For this purpose, Blevins defined a scaling factor termed the “characteristic pressure”, which is a function of the applied loading and the Maximum deflection obtained from the modal solution. The maximum stress can then be calculated using the characteristic pressure and the relation of deflection to stress.
After the stress in a single degree-of-freedom system is derived, a correction to the multi-degrees-of freedom system is required. To this end, Blevins developed a modal correction factor (termed a joint acceptance factor), which characterises the proportion of the random loading that can excite each particular structural mode in the structure. Blevins studied a number of different analytical cases and proposed a modal correction factor of 1.621 for the first mode in the case of a simply supported plate structure.
One drawback of the method proposed by Blevins is that it requires an analytical form of the modal correction factor. Analytical forms for the modal correction factor are only available for a limited range of simple structures (e.g. flat panels). Thus, it will be apparent that the Blevins method can only be applied to a limited range of structures, and cannot be used to model complex structures for use in practical applications.
In view of the impracticalities discussed above, it will be apparent that improved modelling methods are required for practical modelling of the stresses arising in complex structures due to random vibration. Embodiments of the present invention seek to address this need.